Measures of Spread

Range

Formula: $\text{Range} = \text{max} - \text{min}$

Properties:

• Simplest measure
• Uses only two values (min, max)
• Not resistant: one outlier inflates range
• Affected by sample size: larger samples tend to have larger range

Example: data 5, 8, 10, 12, 15 → range = 15 − 5 = 10

Interquartile Range (IQR)

Definitions:

• Q1 (first quartile): 25th percentile
• Q2 (second quartile): 50th percentile = median
• Q3 (third quartile): 75th percentile

Formula: $IQR = Q3 - Q1$

Interpretation: spread of middle 50% of data

Properties:

• Resistant: unaffected by outliers (only uses middle half)
• Paired with median to describe center and spread of skewed data
• Comparable across datasets

Finding Q1 and Q3:

1. Order data
2. Find median (divides into lower and upper halves)
3. Q1 = median of lower half
4. Q3 = median of upper half

Sample Variance (\(s^2\))

Formula: $s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}$

Why n−1? (not n)

• Use n−1 for sample variance (estimating population)
• Corrects for underestimation (Bessel's correction)
• Use n only for population variance \(\sigma^2\)

Properties:

• Measures average squared deviation from mean
• Squared units (e.g., if data in inches, variance in square inches)
• Not resistant: affected by outliers
• Larger dataset → typically larger variance

Sample Standard Deviation (\(s\))

Formula: $s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$

Interpretation:

• Typical (rough average) distance of data points from mean
• Same units as original data
• In approximately normal data: roughly 68% within 1s, 95% within 2s, 99.7% within 3s

Properties:

• Not resistant: affected by outliers
• Paired with mean (for symmetric data)
• Larger s → more spread

Worked Example

Data: Test scores: 60, 70, 75, 80, 85, 90, 95

• n = 7, mean = \(\bar{x} = \frac{595}{7} = 85\)

IQR calculation:

• Lower half: 60, 70, 75 → Q1 = 70
• Upper half: 85, 90, 95 → Q3 = 90
• IQR = 90 − 70 = 20

Variance calculation: $s^2 = \frac{(60-85)^2 + (70-85)^2 + (75-85)^2 + (80-85)^2 + (85-85)^2 + (90-85)^2 + (95-85)^2}{6}$ $= \frac{625 + 225 + 100 + 25 + 0 + 25 + 100}{6} = \frac{1100}{6} \approx 183.33$

Standard deviation: $s = \sqrt{183.33} \approx 13.54$

Interpretation: Scores vary by about 13.54 points from the mean.

Computational Shortcut for \(s^2\)

$s^2 = \frac{\sum x_i^2 - \frac{(\sum x_i)^2}{n}}{n-1}$

Avoids computing \(\bar{x}\) explicitly; useful for calculators or spreadsheets.

Resistance to Outliers

Example: Dataset 5, 8, 10, 12, 15

• Range: 15 − 5 = 10
• IQR: Q1 = 8, Q3 = 12 → IQR = 4
• s ≈ 3.74

Add outlier 100:

• New range: 100 − 5 = 95 (increased 9×)
• New IQR: 8 to 12 (unchanged)
• New s ≈ 37.6 (increased 10×)

Conclusion: IQR is resistant; range and s are not.

Common Mistakes

1. Dividing by n instead of n−1: use n−1 for sample variance
2. Forgetting to square deviations: variance formula uses \((x_i - \bar{x})^2\)
3. Confusing variance and std dev: std dev = \(\sqrt{\text{variance}}\)
4. Misinterpreting s: s is not "maximum distance from mean"; it's typical distance

AP Exam Tip

When comparing spread:

• Symmetric data, no outliers: compare standard deviations
• Skewed data or outliers: compare IQR (resistant)
• Statement: "Distribution A (s = 5) is less spread than Distribution B (s = 8)."

Justify why you chose std dev vs. IQR based on shape of distributions.